R O C Z N IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO Séria I : P R A C E M ATEM AT Y CZN E X I X 1977
T. N oiei (Yatsushiro, Japan)
Almost-bounded sets and some functions
1. Introduction. In 1973, P. Th. Lambrinos [2] introduced the concept of bounded sets in a topological space. Quite recently, in [3] and [4], he has also defined almost-bounded sets and nearly-bounded sets as generalizations of bounded sets and investigated their properties. The relations among these boundedness concepts are similar to those among compactness, almost-compactness and near-eompactness. The main purpose of this note is to show the following two results: 1) The ^-continuous image of an almost-bounded set is almost-bounded; 2) The inverse image of an almost-bounded set under an almost-closed open surjection (not necessarily continuous) with nearly-bounded point inverses is almost- bounded.
Throughout the present note A and T will always denote topological spaces on which no separation axioms are assumed. Let 1 be a subset of a topological space X . The closure of A in X and the interior of A in X are denoted by C1X(A) and Intx (A) respectively. A subset A of X is said to be regularly open if Ink^CL^A)) = A, and regularly dosed il Clx (In t*(^)) = A.
2. Definitions and remarks. The following definitions of boundedness and its generalizations are due to P. Th. Lambrinos [3]. A family SF <= 2X is called an ideal on X if the family = {X — F \ F is a filter on X.
A subfamily of an ideal ^ is called a base (resp. subbase) of W if
= {X — F I F e ^ * } is a base (resp. subbase) of SFù. An ideal 3F is said to be local on a subset A of A if for each point see A, there exists a member FetF such that F is an open set containing so.
D
e f in it io n1. A su b set A of A is said to be bounded (resp. almost- bounded, nearly-bounded) in A [3] if A belongs to ev ery ideal on A h av in g th e follow ing p ro p e rtie s: 1) SF is lo cal on A ; 2) 3F has a b ase (resp.
b ase, su b base) con sistin g of open (resp. closed, regu larly open) sets.
E e m a r k 1. It is known that boundedness => near-boundedness =>
almost-boundedness, but none of these implications is reversible [3].
D e f in it io n 2. A function / : X~>Y is said to have nearly-bounded (resp. bounded) point inverses if for each point yeY, f ~ l (y) is nearly- bounded (resp. bounded) in X .
We shall recall some definitions of functions which are weaker than continuous functions.
D e f in it io n 3. A function / : X-> Y is said to be almost-continuous (resp. в-continuous, weakly-continuous) [10] (resp. [1], [6]) if for each point x e X and each neighborhood V of f(x) in Y, there exists a neigh
borhood V of x in X such that
f(V ) <= Intj.(Clr ( n ) (resp. /(01х (17)) с: С1Г (У), /( 17) c Clr (7)).
R e m a r k 2. It is known that continuity => almost-continuity =>
б-continuity => weak-continuity [8], [10].
D e f in it io n 1. A function / : X-^-Y is said to be almost-open (resp.
almost-closed) [10] if for each regularly open (resp. regularly closed) set A of X ,f(A ) is open (resp. closed) in Y.
R e m a r k 3. Every open (resp. closed) function is almost-open (resp.
almost-closed), but the converse does not hold [10].
3. The O-contimious image of an almost-bounded set. The following lemmas, due to P. Th. Lambrinos, are very usefull in the sequel.
L em m a 1 (Lambrinos [2]). A subset A of X is bounded in X if and only if for any open cover °U of X , there exists a finite subfamily of % such that A <= ( J {U I UeW0}.
L em m a 2 (Lambrinos [3]). A subset A of X is almost-bounded (resp.
nearly-bounded) in X if and only if for any open cover of X , there exists a finite subfamily of °U such that
A = U {01X (P ) I V ' * »} (resp. A c U {In tx (Clx (E0) I Ue1* »})■
P. Th. Lambrinos showed that the continuous image of an almost- bounded set is almost-bounded [3], Theorem 3.1. The word “continuous”
in this result can be replaced by “ 0-continuous” , as the following theorem shows.
T h e o r e m 1. The в-continuous image of an almost-bounded set is almost- bounded.
Proof. Suppose that / : X -> Y is a 0-continuous function and A is an almost-bounded set in X . We shall show that/(A) is an almost-bounded set in Y. For this purpose let [Va | а «г j Y} be any open cover of У. Then for each point xeX , there exists an element a(x) e s# such that f(x) e Va(x).
Since / is 0-continuous, there exists an open neighborhood TJa(x) of x
in X such that f[Glx [Ua(x)))j ClF (Ya(a?)). The family {Ua(x) | xeX )
is an open cover of X . Since A is almost-bounded in X , by Lemma 2,
there exists a finite subfamily ( a ^ ) , a(a?2), ..., a{æn)} of sé such that A c U {C lx (Z7a(a;/) I j = 1? w}. Therefore, we have
By Lemma 2, we observe that f{A ) is almost-bounded in Y.
T h e o r e m 2. The в-continuous almost-open image of a nearly-bounded set is nearly-bounded.
Proof. Suppose th a t/: X -+ Y is a ^-continuous almost-open function and A is a nearly-bounded set in X . We shall show that f(A ) is a nearly- bounded set in Y. For this purpose let [Va | ae sé} be any open cover of Y. Then the family |In tF (ClF (Pa)) | ae is a regularly open cover of Y. Since f is 0-continuous almost-open, it is almost-continuous [6], Theorem 4. Since the inverse image of a regularly open set under an almost-continuous and almost-open function is regularly open [7], Lemma 1, the family | / -1(lntF (01F (Fa))j | a e i j is a regularly open cover of X. Since A is nearly-bounded in X , by Lemma 2, there exists a finite subfamily of stf such that A | / _1(lntF (ClF (Fa))) | ae j >/0J. Thus
By Lemma 2, we observe that f(A ) is nearly-bounded in Y.
C o r o l l a r y 1 (Lambrinos [3]). Let f : X -> Y be a continuous {resp.
continuous open) function. I f A is an almost-bounded {resp. nearly-bounded) set in X , then f(A ) is almost-bounded {resp. nearly-bounded) in Y.
Proof. This is an immediate consequence of Theorem 1 and Theorem 2.
T h e o r e m 3. The almost-continuous {resp. weakly-continuous) image of a bounded set is nearly-bounded {resp. almost-bounded).
Proof. This is proven similarly to Theorem 1.
4. The inverse image of an almost-bounded set.
L em m a 3 (Sikorski [9]). A function f : X ->Y is open if and only if /~1(C1F (B)) <= C1X (/-1(.B)) for every subset В of Y.
T h e o r e m 4. The inverse image of an almost-bounded set under an open and almost-closed surjection with nearly-bounded point inverses is almost-bounded.
Proof. Suppose that f : X -> Y is an open and almost-closed sur
jection with nearly-bounded point inverses. Let В be an almost-bounded set in Y and we will show that f ~ x{B) is an almost-bounded set in X . Let {TJa I ae stf] be any open cover of X . Since / has nearly-bounded point inverses, for each point y e Y, there exists a finite subset {y)
°f ^ such that f~ l{y) c {in tj^ C l^ Z7a)) | ae j/(y)}. Let us put JJy
П П
we have
f{A) c [ l n t F | Cl F ( Y a) | I a e
j/ 0J.
= In t^C U lC M U a) | ae ^V(y)}], then TJ y is a regularly open set containing f ~ l (y). Moreover, put Vy = Y —f ( X — Uy), then we obtain f ~ x(Vy)
<=■ Uy and Vy is an open neighborhood of y in Y because/is almost-closed.
The family {Vy | ye Y) is an open cover of Y. Since В is almost-bounded in Y, by Lemma 2, there exist a finite number of points yt ,y z, ..., yn in Y such* that B <= U{Qlr(^2/y) I j = 1 , 2 , ...,-n}. Since / is open, by using Lemma 3, we obtain
n n n
Г Ч В ) <= и г ‘ (01г ( 7 й ) e и c i ^ O t y ) ) <= U Cl x (Uy,)
3 = 1 3 = 1 3 = 1
= Û U 01x (Ua).
3 = 1 ae s f (yj)
By Lemma 2, we observe that / -1(В) is almost-bounded in X .
Bern ark 4. In Theorem 4, the assumption “open” can be replaced by the following condition: / ~ 1(01F (T)) <= C1X (/_1(Y)) for every open set V of Y.
C orollary 2. Let f : X -> Y be a perfect (closed continuous surjection with compact point inverses) open function. I f A is an almost-bounded set in X (resp. Y), then f (A) (resp. f~ 1(A)) is almost-bounded in Y (resp. X).
Proof. This is an immediate consequence of Theorem 1 and Theorem 4.
T h e o r e m 6. The inverse image of a bounded set under an almost-closed surjection with nearly-bounded point inverses is almost-bounded.
Proof. Suppose that / : X -> Y is an almost-closed surjection with nearly-bounded point inverses. Let В be a bounded set in Y and we will show that / -1(B) is almost-bounded in X . Let { Z7ot
jae j V] be any open cover of X . Similarly to the proof of Theorem 4, for each point yeY, there exist a finite subfamily sV(y) of sV and an open neighborhood Vy of y in Y such that f~ l (Vy) c U {Cl^( Ua) | ae sV(y)} because / is almost- closed and f~ l (y) is nearly-bounded. The family { Vy \ y e Y} is an opkn cover of Y. Since В is a bounded set in Y, by Lemma 1, there exist a finite number of point 2 /x, y2, . . . , y nm Y such that B c ( J { Vyi \ j = 1, 2, ..., n}.
Therefore, we obtain
П П
m ^ u m ) c u и oix (u a).
j = l 3 = 1 aes/(yj)
By Lemma 2, we observe that / _1(B) is almost-bounded in X .
T h e o r e m 6. The inverse image of a bounded set under a closed surjection with nearly-bounded (resp. bounded) point inverses is nearly-bounded (resp- bounded).
Proof. This is proven similarly to Theorem 4.
References
[1] S. V. Fom in , Extensions of topological spaces, Ann. of Math. 44 (1943), p. 471- 480.
[2] P. Th. L am b rin o s, A topological notion of boundedness, Manuscripta Math.
10 (1973), p. 289-296.
[3] — Some weaker forms of topological boundedness (unpublished).
[4] — Generalized boundedness topological concepts, Notices Amer. Math. Soc. 21 (1974), p. A-504.
[6] N. L ev in e, A decomposition of continuity in topological spaces, Amer. Math.
Monthly 68 (1961), p. 44-46.
[6] T. N oiri, Between continuity and weak-continuity, Boll. Un. Mat. Ital. (4) 9 (1974), p. 647-654.
[7] — A remark on almost-continuous mappings, Proc. Japan Acad. 50 (1974), p. 205-207.
[8] P. J . Papp, Almost-continuous implies в-continuous, Notices Amer. Math. Soc.
20 (1973), p. A-175.
[9] R. S ik o rsk i, Closure homomorphisms and interior mappings, Fund. Math. 41 (1955), p. 12-20.
[10] M. K. S in g a l and A sh a }üani S in g al, Almost-continuous mappings, Yoko hama Math. J . 16 (1968), p. 63-73.
Y A TSU SH IR O C O L L E G E OF T E C H N O LO G Y , Y A T S U SH IR O , JA P A N
